-- Main.ebs22 - 2016-01-07 -- Main.srs74 - 2015-12-22

Pseudomanifold Triangulations on 10 Vertices

Complex: 10_a_b_c_d_0_e_0_f_0_g

• a= number of vertex links homeomorphic to the sphere
• b= number of vertex links homeomorphic to the real projective plane
• c= number of vertex links homeomorphic to the torus
• d= number of vertex links homeomorphic to the Klein bottle
• e= number of vertex links homeomorphic to the genus three nonorientable surface
• f= number of vertex links homeomorphic to the genus four nonorientable surface
• g= number of vertex links homeomorphic to the genus five nonorientable surface

χ - Euler characteristic of all the complexes on 10 vertices with the given vertex links.

Triangulations - The number of triangulations with the given vertex links on 10 vertices. Clicking on the link gives all of the triangulations.

minG2 - The minimum g₂ over all triangulations with the given vertex links on 10 vertices. Clicking the link gives a list of the complexes which realize the minimum g₂.

H₁, H₂, H₃ - Integer homology groups of all of the triangulations with the given vertex links on 10 vertices. The homology group is trivial if blank. For H₂ the shorthand n,[2] stands for the direct sum of Z/2Z and the free abelian group of rank n.

Γ - Γ is the minimum of g₂ over all triangulations of a three-dimensional normal pseudomanifold with the given singular vertices. A letter in this column indicates that minG2=Γ and the proof is indicated below. A superscript ' indicates that Γ=minG2-1 and 11 vertices are needed to realize Γ.

• a - For any vertex v of Δ, g₂(Δ) ≥ g₂ (link v).
• b - If n is the number of singular vertices, then g₂ ≥ 2 χ - ( n-3 choose 3). If n-3 < 3, then the binomial coefficient is interepreted as zero.
• c - If Δ has 8 singular vertices and m of them are Klein bottles, then g₂ ≥ 2 χ - 10 + (m/3)
• d - If Δ has 8 singular vertices and any of them are real projective planes, then g₂ ≥ 2 χ - 7
• e - If Δ has 8 singular vertices including 3 projective planes and 2 Klein bottles, then g₂ ≥ 2 χ - 5

f-vector - A nonempty entry indicates that all possible f-vectors for complexes with the given singular vertices is known.

Except where otherwise noted, the f-vectors are characterized through h- and g-vectors by, h₀=1, h₄=1-χ, h₃ - h₁ = 2 χ, h₁ ≥ f₀-4, and Γ ≤ g₂ ≤ (g₁ +1 choose 2), where f₀ is the minimum number of vertices required for a complex with the given singularities.

• The first entry is the minimum number of vertices possible for the given singularities
• 10 indicates that the possible f-vectors for PL-homeomorphic complexes for every complex in the group are the same and equal all possible f-vectors for that particular group of singularities
• 10, # indicates that the possible f-vectors of complexes PL-homeomorphic to complex # equals all possible f-vectors for that group of singularities.
• 10, #, β There is no complex with g-vector (5, Γ) for these singularities.
• 9, # indicates that the possible f-vectors of complexes PL-homeomorphic to complex # at http://www.math.cornell.edu/~takhmejanov/pseudoManifolds.html with the same singularities equals all possible f-vectors for that group of singularities.
• 9, #1, α There is no complex with g-vector (4,6) for these singularities.
• 8, N# indicates that the possible f-vectors of complexes PL-homeomorphic to complex N# in "Three-Dimensional Pseudomanifolds on Eight Vertices", B. Datta and N. Nilakantan, Indian J. of Mathematics and Mathematical Sciences, 2008, equals all possible f-vectors for that group of singularities.
• 7, The one-vertex suspension of the six-vertex triangulation of the real projective plane can be used to prove that the f-vectors of the suspension of the real projective plane has the same f-vectors as all complexes with exactly two singular vertices each with link homeomorphic to the real projective plane.
• 5, f-vectors of three-manifolds equal all possible f-vectors of S³.

≅ - * indicates that all complexes in this row are known to be PL-homeomorphic.